A linear motion problem that leads to the harmonic formula.

A car travels with a speed of 40 miles per hour for the first half of the way. Then, the car travels with a speed of 60 miles per hour for the second half of the way. What is the average speed?

Average speed =

total distance/total time

First notice that it is not possible to use directly the speed formula since we do not know for how long the car kept driving with a speed of 40 m/h and then 60 m/h. However, with some manipulation, we can still tackle the problem.

Let t1 be the time it took to travel the first half of the total distance

Let d be the first half of the total distance.

t1 =

d/40

Let t2 be the time it took to travel the second half of the total distance

Let d be the second half of the total distance.

t2 =

d/60

Total time = t1 + t2 = d/40 + d/60

Total distance = d + d = 2d

Now replace these in the formula

Average speed =

total distance/total time

Average speed =

2d/d/40 + d/60

Average speed =

2d/d(1/40 + 1/60)

Cancel d and the average speed =

2/(1/40 + 1/60)

Now, you can see that it looks like we are calculating the harmonic mean for 2 numbers by using the formula above.